Properties of the Lindemann Mechanism in Phase Space

TL;DR

This paper analyzes the Lindemann mechanism in phase space, proving stability, uniqueness of solutions, and asymptotic behaviors, thereby deepening understanding of its dynamical properties in unimolecular decay models.

Contribution

It provides a comprehensive phase space analysis of the Lindemann mechanism, including stability, solution uniqueness, and asymptotic behavior, which were not fully characterized before.

Findings

The origin is globally asymptotically stable.

There is a unique scalar solution (slow manifold).

All solutions exhibit specific asymptotic behaviors.

Abstract

We study the planar and scalar reductions of the nonlinear Lindemann mechanism of unimolecular decay. First, we establish that the origin, a degenerate critical point, is globally asymptotically stable. Second, we prove there is a unique scalar solution (the slow manifold) between the horizontal and vertical isoclines. Third, we determine the concavity of all scalar solutions in the nonnegative quadrant. Fourth, we establish that each scalar solution is a centre manifold at the origin given by a Taylor series. Moreover, we develop the leading-order behaviour of all planar solutions as time tends to infinity. Finally, we determine the asymptotic behaviour of the slow manifold at infinity by showing that it is a unique centre manifold for a fixed point at infinity.